Volatility skew tells us that options with the same maturity at different strikes can have different implied vol. However, can a corresponding call and put for the same strike and maturity have different implied vol?

3 Answers
3

Taking away all frictions and incomplentess of the market, the theory says that European Call and Puts do have the same implied volatility unless there is an arbitrage opportunity by put call parity
$$
C(t,K) - P(t,K) = DF_t(F_t - K)\ .
$$
If you plug the Black-Scholes formula here for the prices of the call and the put, you will see that the equality only holds if and only if volatilities are equal.
$$
DF_t[F_t(\Phi(d_+^{Call}) + \Phi(-d_+^{Put})) - K(\Phi(d_-^{Call}) + \Phi(-d_-^{Put}))] = DF_t(F_t-K)
$$
Since $\Phi(x)+\Phi(-x)=1$, put call parity holds if and only if $d_\pm^{Call} = d_\pm^{Put}$, so if and only if $\sigma_{Call}(t,K) = \sigma_{Put}(t,K)$.

However, in practice there are bid-ask spreads and liquidity issues which implies that observable prices of European options do no align necessarily to the theory.

However, for American options (the standard options traded on Equity stocks) we can still think in terms of implied volatility but there is no such thing as a put-call parity so implied volatilities are not necessarily equal anymore. There are some put-call parity style inequalities but those are not strong enough to guarantee the equality of volatilities.

Implied volatility does not have to be equal (so yes, it can be different) for a call and put of same underlying, underlying borrow rates, time to expiration, strike if:

If the underlying is a stock and the underlying cannot be easily borrowed for short selling

If there are dividends or other costs of carry involved

If there is not unlimited liquidity in the market

In the absence of market turbulence.

In the absence of those such call and put should have matching implied volatility, under Put-Call parity. Please keep in mind above conditions can be met much more often than most academic papers suggest. I can name you multiple examples for each above mentioned points that occurred just over the past 10 years that may have pushed the call and put implied vols significantly out of whack, sometimes for short periods of time, sometimes longer periods.

First: what you use in the call or put formula is volatility of underlying; it is the same underlying, so volatility implied by call and put has to be the same. It is vol of underlying asset.

Remember put-call parity

$call-put=S-e^{-rt}K$

$call=put+S-e^{-rt}K$ by a pure arbitrage rule

This means that volatility of call, as variable equal to put+constant, is the same as volatility of put. Since only volatility induced to both of these comes from volatiity of the asset, I am sure it can be shown that this volatilty of asset must be the same for call and put.

I do not follow your reasoning at all. Invoking put-call parity does not lead one to say that the "volatility of call...is the same as volatility of put", as put-call parity can be derived using model-free arguments only (i.e., without using the Black-Scholes model, etc.).
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wswMar 26 '13 at 3:19

no, if the volatility is not the same, then you can use put-call parity to profit from this fact. and put-call parity is also derived from Black-Scholes. this model as any else has to assure that put-call parity holds
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AB_Mar 26 '13 at 6:41

@chrisaycock also this is central element of quoting volatility surface: you are using put-call parity transformed to delta relationship. this is basic concept: why you don't have two surfaces, one for put and one for call? : p
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AB_Mar 26 '13 at 11:27

@cf16 What about what? What are you talking about? Why are you pinging me? I edited your answer; did you mean to ask William?
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chrisaycockMar 26 '13 at 11:29